In magnetic resonance (MR) imaging, signal values are acquired as a function of time and changes in the magnetic field. The conventional detection hardware outputs two signals, which are digitized synchronously and independently. The two numbers that represent the signal at a given instant are stored as the real and imaginary part of a complex number. These complex numbers are arranged in a two-dimensional or higher-dimensional matrix of complex numbers referred to as raw data or k-space data. This matrix is a “Fourier hologram,” which is reconstructed to form a useful image. For ordinary reconstruction, an “inverse” Fourier transformation is performed on each row of data (the “row transform”) to form a “half-transformed” matrix. The inverse Fourier transform is then performed on each column of the half-transformed matrix (the “column transform”) to form a complex image. The combined effect of these two transformations is referred to as the 2-dimensional (2-D) Fourier transform. For simplification purposes, 2-D reconstructions are described herein, but the principles may be applied to higher dimensions in a non-limiting fashion. In the most general case of Fourier reconstruction, a complex MR image having N rows and M columns, and comprising NM pixel values, each with independent real and imaginary values, or equivalently magnitudes and phases, is the result of Fourier transformations of a matrix comprising NM complex raw data points. Thus, both the raw data matrix and the complex image have 2NM independent parameters.
The radiofrequency (RF) excitation pulses that rotate the nuclear magnets (spins) from their equilibrium magnetization parallel to the main magnetic field (conventionally termed the +z direction) into the x-y transverse plane, leave all or most of the spins pointing in one direction (e.g., they direction). If all of the spins have the same direction, they have the same phase, and all of the complex pixels of the reconstructed image also have the same phase. In this case, the number of independent parameters in the image is halved to NM+1, and it is possible to reconstruct the image from approximately half of the NM complex raw data points without sacrificing image quality, other than increasing the noise in the image (reducing the “signal-to-noise ratio”). This reduction in the number of data points may be used, for example, to shorten the scan time, which increases patient comfort and reduces the artifacts created by patient motion.